Title: On the structure of
cofree Hopf algebras
Abstract: We prove an
analogue of the Poincaré-Birkhoff-Witt theorem and
of the
Milnor-Moore theorem for non-cocommutative Hopf algebras. The
primitive part
of a cofree Hopf algebra is a B-infinity-algebra. We
construct a
universal enveloping functor U2 from B-infinity-algebras to 2-
associative
algebras, i.e. algebras equipped with two associative
operations. We
show that any cofree Hopf algebra H is of the form U2(Prim H).